26 PHYWE Systeme GmbH & Co. KG · D-37070 GöttingenLaboratory Experiments Physics

Mechanics Dynamics

1.3.11-00 Projectile motion

Principle:A steel ball is fired by a spring at dif-ferent velocities and at different angles to the horizontal. The rela-tionships between the range, theheight of projection, the angle of in-clination, and the firing velocity are determined.

Maximum range as a function of the angle of inclina tion � for different initialvelocity v0: Curve 1 v0 = 5.3 m/s

Curve 2 v0 = 4.1 m/sCurve 3 v0 = 3.1 m/s

Tasks:1. To determine the range as a func-

tion of the angle of inclination.

2. To determine the maximum heightof projection as a function of theangle of inclination.

3. To determine the (maximum)range as a function of the ini tialvelocity.

What you can learn about …

� Trajectory parabola� Motion involving uniform

acceleration� Ballistics

Ballistic unit 11229.10 1

Recording paper, 1 roll, 25 m 11221.01 1

Steel ball, hardened and polished, d = 19 mm 02502.01 2

Two tier platform support 02076.03 1

Meter Scale, l = 1000 x 27 mm 03001.00 1

Barrel base -PASS- 02006.55 1

Speed measuring attachment 11229.30 1

Power supply 5 VDC/2.4 A with DC-socket 2.1 mm 13900.99 1

What you need:

Complete Equipment Set, Manual on CD-ROM includedProjectile motion P2131100

LEP1.3.11

-00Projectile motion

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P2131100 1

Related topicsTrajectory parabola, motion involving uniform acceleration,ballistics.

PrincipleA steel ball is fired by a spring at different velocities and at dif-ferent angles to the horizontal. The relationships between therange, the height of projection, the angle of inclination, and thefiring velocity are determined.

EquipmentBallistic unit 11229.10 1Recording paper, 1 roll, 25 m 11221.01 1Steel ball, d = 19 mm 02502.01 2Two-tier platform support 02076.03 1Meter scale, demo, l = 1000 mm 03001.00 1Barrel base 02006.10 1Speed measuring attachement 11229.30 1Power supply 5 VDC/2.4 A 13900.99 1

Tasks1. To determine the range as a function of the angle of incli-

nation.

2. To determine the maximum height of projection as a func-tion of the angle of inclination.

3. To determine the (maximum) range as a function of the ini-tial velocity.

Set-up and procedureThe ballistic unit is adjusted. The scale is set to read 90° anda ball is fired upwards (setting 3) and is caught in the hand.The support base adjusting screws are turned until a verticalprojection is obtained.The initial velocities of the ball corresponding to the three ten-sion stages of the firing spring can be determined using thespeed measuring attachment, or from the maximum height fora vertical projection from the expression v0 = . The ini-tal velocities may vary greatly from unit to unit.The 2-tier platform support (02076.01) is used for determiningthe range. To mark the points of impact, the recording strip issecured to the bench with adhesive tape. It is best to measurethe long ranges before the short ones (secondary impactpoints!) and to mark the primary impact points with a felt pen.The distance from the ballistic unit is frequently checked withthe meter scale during the test. An empty box can be placedbehind the bench to catch the balls.

22gh

Fig. 1: Experimental set-up for measuring the maximum range of a projectile with additional equipment to measure the initialvelocity.

LEP1.3.11

-00Projectile motion

P2131100 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen2

To measure the height of projection the meter scale is clam-ped in the barrel base and moved parallel to the plane of pro-jection. The empty box is again used to catch the balls. Theheights of projection can be determined ballistically quite wellby eye.

Theory and evaluationIf a body of mass m moves in a constant gravitational field(gravitational force mg

�), the motion lies in a plane.

If the coordinate system is laid in this plane (x, y plane – seefig. 2) and the equation of motion:

where:

r�

= (x,y) ; g�

= (0, –g)

is solved, then, with the initial conditions

r (0) = 0

v�

(0) = (v0 cos �, v0 sin �)

we obtain the coordinates as a function of time t:

From this, the maximum height of projection h is obtained asa function of the angle of projection �:

and the maximum range s is:

From the regression line of the data of fig. 5, using the expres-sion:

Y = A · X B

we obtain the exponent

B = 2.01 = 0.001

s �v0

2

g sin 2 f

h �v0

2

2g sin2 f

y 1t 2 � v0 · sinf · t �g

2 t2 :

x 1t 2 � v0 · cosf · t

m d2

dt2 r S1t 2 � m g S

Fig. 2: Movement of a mass point under the effect of gravita-tional force.

Fig. 3: Maximum range as a function of the angle of inclination� for different initial velocity v0:

Curve 1 v0 = 5.3 m/sCurve 2 v0 = 4.1 m/sCurve 3 v0 = 3.1 m/s

Fig. 4: Maximum height of projection h as a function of theangle of inclination � for the initial velocities as in Fig. 1.

LEP1.3.11

-00Projectile motion

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P2131100 3

NoteTo ensure an accurate determination of the initial velocity, thetime taken for the ball to cover the measuring distance mustbe taken into account. If vexp is the experimentally determinedinital velocity we obtain

where d is the distance between the point of rotation and thecentre between the light barriers.

v0 � 2vexp2 � 2g d sin f

Fig. 5: Maximum range s as a function of the inital velocity vowith a fixed angle of inclination � = 45°.

LEP1.3.11

-00Projectile motion

P2131100 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen4